Decentralized Distributed Graph Coloring II: degree+1-Coloring Virtual Graphs

TL;DR

This paper introduces a nearly optimal distributed algorithm for degree+1 coloring of virtual graphs embedded in communication networks, generalizing classical graph coloring to new settings with constant congestion.

Contribution

It provides the first general framework for coloring virtual graphs in distributed systems, extending classical methods and analyzing complexity based on embedding congestion.

Findings

Nearly logarithmic double-logarithmic round complexity for coloring virtual graphs.

Coloring virtual graphs with constant congestion is as efficient as coloring ordinary graphs.

The approach generalizes classical distributed coloring to broader virtual graph settings.

Abstract

Graph coloring is fundamental to distributed computing. We give the first general treatment of the coloring of virtual graphs, where the graph $H$ to be colored is locally embedded within the communication graph $G$. Besides generalizing classical distributed graph coloring (where $H=G$), this captures other previously studied settings, including cluster graphs and power graphs. We find that the complexity of coloring a virtual graph depends on the edge congestion of its embedding. The main question of interest is how fast we can color virtual graphs of constant congestion. We find that, surprisingly, these graphs can be colored nearly as fast as ordinary graphs. Namely, we give a $O(\log^4\log n)$-round algorithm for the deg+1-coloring problem, where each node is assigned more colors than its degree. This can be viewed as a case where a distributed graph problem can be solved even when the operation of each node is decentralized.